Subgroups — C₉×#3061 #11767782

Order 1152.

726 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#537	C₂	1	2	yes	maximal, complemented
4	C₉×#468	C₂	1	4	yes	maximal, complemented
5	C₉×#467	C₂	1	2	yes	maximal, complemented
6	C₉×#466	C₂	1	2	yes	maximal, complemented
7	C₂×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
8	C₉×#417	C₂	1	2	yes	maximal, complemented
9	C₃×#3061	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	C₂²	1	2	yes	—
11	C₂×C₉×#41	C₂²	1	1	yes	characteristic
12	C₉×#135	—	2	4	no	—
13	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
16	C₉×#131	—	2	4	no	—
17	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₉×M₁₆	C₂²	1	2	yes	—
19	C₂²×C₈×C₉	C₂²	1	2	yes	—
20	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
21	C₉×C₂²⋊Q₈	—	2	2	no	—
22	C₉×C₂²⋊Q₈	C₂²	1	4	yes	—
23	C₉×#121	—	2	2	no	—
24	C₉×#121	—	2	4	no	—
25	C₉×#121	C₂²	1	4	yes	—
26	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
27	C₉×#107	C₂²	1	2	yes	—
28	C₉×#107	C₂²	1	2	yes	—
29	C₉×#103	C₂²	1	4	yes	—
30	C₉×#102	C₂²	1	4	yes	—
31	C₂×C₃×#135	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
33	C₃×#537	C₂×C₃	1	2	yes	—
34	C₃×#468	C₂×C₃	1	4	yes	—
35	C₃×#467	C₂×C₃	1	2	yes	—
36	C₃×#466	C₂×C₃	1	2	yes	—
37	C₂×C₃×#107	C₂×C₃	1	1	yes	characteristic
38	C₃×#417	C₂×C₃	1	2	yes	—
39	C₉×#41	—	2	2	no	—
40	C₉×#41	—	2	2	no	—
41	C₉×#41	—	2	4	no	—
42	C₉×#41	—	2	4	no	—
43	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₉	C₂³	1	2	yes	—
45	C₂×D₄×C₉	—	2	2	no	—
46	C₂×D₄×C₉	—	2	4	no	—
47	C₂×D₄×C₉	D₄	1	2	yes	—
48	C₂×D₄×C₉	D₄	1	1	yes	characteristic
49	C₂×D₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	—	2	2	no	—
51	C₂²×C₄×C₉	—	2	2	no	—
52	C₂²×C₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
54	C₉×Q₁₆	—	2	2	no	—
55	C₉×SD₁₆	—	2	4	no	—
56	C₉×D₈	—	2	2	no	—
57	C₉×M₁₆	—	2	2	no	—
58	C₉×M₁₆	—	2	4	no	—
59	C₂×C₈×C₉	—	2	4	no	—
60	C₂×C₈×C₉	D₄	1	4	yes	complemented
61	C₂×C₈×C₉	C₂³	1	2	yes	—
62	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
64	C₉×C₄⋊C₄	—	2	2	no	—
65	C₉×C₄⋊C₄	—	2	4	no	—
66	C₉×C₄⋊C₄	C₂³	1	4	yes	—
67	C₉×C₂²⋊C₄	—	2	4	no	—
68	C₉×C₂²⋊C₄	—	2	4	no	—
69	#3061	C₉	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
71	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
72	C₃×#135	—	2	4	no	—
73	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
75	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
76	C₃×#131	—	2	4	no	—
77	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
79	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
80	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
81	C₃×C₂²⋊Q₈	—	2	2	no	—
82	C₃×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
83	C₃×#121	—	2	2	no	—
84	C₃×#121	—	2	4	no	—
85	C₃×#121	C₂²×C₃	1	4	yes	—
86	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₃×#107	C₂²×C₃	1	2	yes	—
88	C₃×#107	C₂²×C₃	1	2	yes	—
89	C₃×#103	C₂²×C₃	1	4	yes	—
90	C₃×#102	C₂²×C₃	1	4	yes	—
91	C₂³×C₉	—	2	2	no	—
92	C₂³×C₉	C₂×D₄	1	2	yes	—
93	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
94	Q₈×C₉	—	2	2	no	—
95	Q₈×C₉	—	2	2	no	—
96	Q₈×C₉	—	2	2	no	—
97	D₄×C₉	—	2	4	no	—
98	D₄×C₉	—	2	2	no	—
99	D₄×C₉	—	2	2	no	—
100	D₄×C₉	—	2	2	no	—
101	D₄×C₉	—	4	4	no	—
102	D₄×C₉	—	8	4	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	4	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	4	4	no	—
108	C₂×C₄×C₉	—	8	2	no	—
109	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₉	#41	1	2	yes	—
113	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
114	C₈×C₉	—	2	2	no	—
115	C₈×C₉	C₂×D₄	1	2	yes	—
116	C₈×C₉	C₂×D₄	1	2	yes	—
117	C₂×#135	C₂×C₉	1	1	yes	characteristic, complemented
118	C₂×#131	C₂×C₉	1	1	yes	characteristic, complemented
119	#537	C₂×C₉	1	2	yes	complemented
120	#468	C₂×C₉	1	4	yes	complemented
121	#467	C₂×C₉	1	2	yes	complemented
122	#466	C₂×C₉	1	2	yes	complemented
123	C₂×#107	C₂×C₉	1	1	yes	characteristic, complemented
124	#417	C₂×C₉	1	2	yes	complemented
125	C₃×#41	—	2	2	no	—
126	C₃×#41	—	2	2	no	—
127	C₃×#41	—	2	4	no	—
128	C₃×#41	—	2	4	no	—
129	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
130	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	—	2	4	no	—
133	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
134	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
135	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₃×C₄	—	2	2	no	—
137	C₂²×C₃×C₄	—	2	2	no	—
138	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
139	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
140	C₃×Q₁₆	—	2	2	no	—
141	C₃×SD₁₆	—	2	4	no	—
142	C₃×D₈	—	2	2	no	—
143	C₃×M₁₆	—	2	2	no	—
144	C₃×M₁₆	—	2	4	no	—
145	C₂×C₃×C₈	—	2	4	no	—
146	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
147	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
148	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	—	2	4	no	—
152	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
153	C₃×C₂²⋊C₄	—	2	4	no	—
154	C₃×C₂²⋊C₄	—	2	4	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	4	2	no	—
161	C₂²×C₉	—	4	2	no	—
162	C₂²×C₉	—	4	2	no	—
163	C₂²×C₉	#121	1	2	yes	—
164	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
165	C₄×C₉	—	2	2	no	—
166	C₄×C₉	—	4	2	no	—
167	C₄×C₉	—	4	4	no	—
168	C₄×C₉	#121	1	2	yes	—
169	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
170	C₄×C₉	C₂×#41	1	1	yes	characteristic
171	C₂×#41	C₂²×C₉	1	2	yes	—
172	C₂×#41	C₂²×C₉	1	1	yes	characteristic
173	#135	—	2	4	no	—
174	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
175	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
176	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
177	#131	—	2	4	no	—
178	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
179	C₂×M₁₆	C₂²×C₉	1	2	yes	—
180	C₂²×C₈	C₂²×C₉	1	2	yes	—
181	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	2	no	—
183	C₂²⋊Q₈	C₂²×C₉	1	4	yes	—
184	#121	—	2	2	no	—
185	#121	—	2	4	no	—
186	#121	C₂²×C₉	1	4	yes	—
187	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
188	#107	C₂²×C₉	1	2	yes	—
189	#107	C₂²×C₉	1	2	yes	—
190	#103	C₂²×C₉	1	4	yes	—
191	#102	C₂²×C₉	1	4	yes	—
192	C₂³×C₃	—	2	2	no	—
193	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
194	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
195	C₃×Q₈	—	2	2	no	—
196	C₃×Q₈	—	2	2	no	—
197	C₃×Q₈	—	2	2	no	—
198	C₃×D₄	—	2	4	no	—
199	C₃×D₄	—	2	2	no	—
200	C₃×D₄	—	2	2	no	—
201	C₃×D₄	—	2	2	no	—
202	C₃×D₄	—	4	4	no	—
203	C₃×D₄	—	8	4	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	4	4	no	—
209	C₂×C₃×C₄	—	8	2	no	—
210	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
211	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₃×C₄	C₃×#41	1	2	yes	—
214	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
217	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
218	C₂×C₉	—	2	2	no	—
219	C₂×C₉	—	4	2	no	—
220	C₂×C₉	—	8	2	no	—
221	C₂×C₉	C₂×#121	1	1	yes	characteristic
222	C₂×C₉	#576	1	1	yes	characteristic
223	C₂×C₉	#578	1	1	yes	characteristic
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	#41	—	2	4	no	—
227	#41	—	2	4	no	—
228	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
229	C₂×Q₈	C₂³×C₉	1	2	yes	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	D₄×C₉	1	2	yes	—
233	C₂×D₄	D₄×C₉	1	1	yes	characteristic
234	C₂×D₄	C₂³×C₉	1	2	yes	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂³×C₉	1	2	yes	—
238	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
239	Q₁₆	—	2	2	no	—
240	SD₁₆	—	2	4	no	—
241	D₈	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	D₄×C₉	1	4	yes	complemented
246	C₂×C₈	C₂³×C₉	1	2	yes	—
247	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
248	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	C₂³×C₉	1	4	yes	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	4	2	no	—
260	C₂²×C₃	—	4	2	no	—
261	C₂²×C₃	—	4	2	no	—
262	C₂²×C₃	C₃×#121	1	2	yes	—
263	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	—	4	2	no	—
267	C₃×C₄	C₃×#121	1	2	yes	—
268	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
269	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
270	C₉	#3061	1	1	yes	characteristic, direct product factor, Sylow
271	C₂³	—	2	2	no	—
272	C₂³	C₂×D₄×C₉	1	2	yes	—
273	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
274	Q₈	—	2	2	no	—
275	Q₈	—	2	2	no	—
276	Q₈	—	2	2	no	—
277	D₄	—	2	4	no	—
278	D₄	—	2	2	no	—
279	D₄	—	2	2	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	8	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
290	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
292	C₂×C₄	C₉×#41	1	2	yes	—
293	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	C₂×D₄×C₉	1	2	yes	—
296	C₈	C₂×D₄×C₉	1	2	yes	—
297	C₂×C₃	—	2	2	no	—
298	C₂×C₃	—	4	2	no	—
299	C₂×C₃	—	8	2	no	—
300	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
301	C₂×C₃	C₃×#576	1	1	yes	characteristic
302	C₂×C₃	C₃×#578	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	C₉×#121	1	2	yes	—
312	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	4	no	—
316	C₄	C₉×#121	1	2	yes	—
317	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
318	C₄	C₂×C₉×#41	1	1	yes	characteristic
319	C₃	C₃×#3061	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	4	2	no	—
322	C₂	—	8	2	no	—
323	C₂	C₂×C₉×#121	1	1	yes	characteristic
324	C₂	C₉×#576	1	1	yes	characteristic
325	C₂	C₉×#578	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#3061	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₉×#41 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₉×#135	C₂	semidirect
2. C₂×C₉×#131	C₂	semidirect
3. C₉×#537	C₂	semidirect
4. C₉×#468	C₂	semidirect
5. C₉×#467	C₂	semidirect
6. C₉×#466	C₂	semidirect
7. C₂×C₉×#107	C₂	semidirect
8. C₉×#417	C₂	semidirect
60. C₂×C₈×C₉	D₄	semidirect
69. #3061	C₉	direct (N × H)
117. C₂×#135	C₂×C₉	semidirect
118. C₂×#131	C₂×C₉	semidirect
119. #537	C₂×C₉	semidirect
120. #468	C₂×C₉	semidirect
121. #467	C₂×C₉	semidirect
122. #466	C₂×C₉	semidirect
123. C₂×#107	C₂×C₉	semidirect
124. #417	C₂×C₉	semidirect
245. C₂×C₈	D₄×C₉	semidirect
270. C₉	#3061	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₉×#135	2
C₂×C₉×#131	2
C₉×#537	2
C₉×#468	2
C₉×#467	2
C₉×#466	2
C₂×C₉×#107	2
C₉×#417	2
C₃×#3061	3

Subgroups — C9×#3061 #11767782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3061 #11767782